Spherical elastomeric bearing with improved shim thickness

ABSTRACT

An elastomeric spherical bearing includes a multiple of shims, at least two of which have different thicknesses. In one exemplary embodiment, each of the shims has a different thickness with a generally equivalent stress on each shim.

BACKGROUND

The present invention relates to an elastomeric spherical bearing.

One goal of elastomeric spherical bearing design is thesmallest/lightest package that meets the desired design liferequirements. Conventional spherical elastomeric bearing optimizationprocedures have only optimized the elastomeric layers and incorporatedsingle thickness non-extensible layers such as metal shims throughoutthe entire bearing.

SUMMARY

An elastomeric spherical bearing according to an exemplary aspect of thepresent invention includes a multitude of shims, each of said multitudeof shims mounted between at least two of the multiple of elastomericlayers, a first shim of the multitude of shims having a first thicknessand a second shim of the multitude of shims having a second thickness,the second thickness different than the first thickness.

A method of calculating a nonresilient shim thickness for a sphericalelastomeric bearing according to an exemplary aspect of the presentinvention includes Determining a first thickness of a first of amultitude of nonresilient shim layers to satisfies a first set ofcriteria. Determining a second thickness of a second of the multitude ofnonresilient shim layers outboard of the first of the multitude ofnonresilient shim layers to satisfies a second set of criteria, thesecond thickness different than the first thickness.

A method of calculating a nonresilient shim layer thickness for aspherical elastomeric bearing according to an exemplary aspect of thepresent invention includes determining a first shim thickness of a firstof a multitude of nonresilient shim layers in the spherical elastomericbearing, each of the multitude of nonresilient shim layers sandwichedbetween a first pair of elastomeric layers. Determining a elastomericlayer thickness for the first pair of elastomeric layers. Modifying thefirst shim thickness and the elastomeric layer thickness in response toan addition of a second a of the multitude of nonresilient shim layersand a second pair of elastomeric layers outboard of the first of themultitude of nonresilient shim layers and the first pair of elastomericlayers, the modifying continuing until each of the multitude ofnonresilient shim layers have a generally equivalent stress.

BRIEF DESCRIPTION OF THE DRAWINGS

The various features and advantages of this invention will becomeapparent to those skilled in the art from the following detaileddescription of the currently disclosed embodiment. The drawings thataccompany the detailed description can be briefly described as follows:

FIG. 1 is a perspective view a rotor head assembly utilizing anelastomeric bearing according to one non-limiting embodiment of thepresent invention;

FIG. 2A is an enlarged broken-away perspective view of the elastomericbearing in combination with a rotor assembly yoke and shear segment ofthe rotor hub assembly of FIG. 1;

FIG. 2B depicts an enlarged view of the elastomeric laminates of theelastomeric bearing of FIG. 2A;

FIG. 3 is a schematic sectional view of a spherical elastomeric layer ofthe elastomeric bearing design envelope;

FIG. 4 is a schematic sectional view of a spherical elastomeric bearingwith details of one inner elastomeric layer, one outer elastomeric layerand a nonresilient shim therebetween;

FIG. 5 is an expanded schematic sectional view of differential pressureon an inner surface and an outer surface of a nonresilient shim;

FIG. 6A is an expanded schematic sectional view of an undeformednonresilient shim;

FIG. 6B is an expanded schematic sectional view of the nonresilient shimof FIG. 6A in a deformed condition to illustrate the deformationmechanism which generates the primary shim stress;

FIG. 7A is a flow chart illustrating a calculation procedure forcalculation of Compressive Load T_(c) And Torsional Motion (θ) accordingto one non-limiting embodiment;

FIG. 7B is a flow chart illustrating a calculation procedure forcalculation of Compressive Load (T_(c)); Torsional Motion (θ); AndCocking Motions (β) according to one non-limiting embodiment;

FIG. 8 is a graph representing a shim thickness relative a shim numberfor one elastomeric bearing with three shim layers according to onenon-limiting embodiment of the present invention; and

FIG. 9 is a graph of the elastomeric bearing of FIG. 3 illustrating anessentially equivalent shim stress at each of the three shim layers.

DETAILED DESCRIPTION OF THE DISCLOSED EMBODIMENT

Referring to FIG. 1, a rotor hub assembly 10 typical of a rotary-wingaircraft includes a hub retention member 12 which drives a multitude ofrotor blade assemblies 14 about an axis of rotation 16.

The hub retention member 12 includes a multitude of radial spokes 20 andshear segments 22. Each shear segment 22, in combination with itsrespective radial spokes 20, form a structural loop for accepting arotor assembly yoke 24. The yoke 24 is generally C-shaped andcircumscribes, in looped fashion, the respective shear segment 22. Theyoke 24 is disposed in combination with a cuff structures 28 which, inturn, mount to the root end of each rotor blade assembly 14.

A spherical elastomeric bearing assembly 30 is interposed between eachrotor assembly yoke 24 and the respective shear segment 22 toaccommodate the multi-directional rotation of the rotor blade assembly14.

Referring to FIG. 2A, the spherical elastomeric bearing 30 is shown incombination with a rotor assembly yoke 24 and a respective shear segment22. It should be understood that although a particular rotor hubapplication is illustrated in the disclosed non-limiting embodiment,elastomeric bearing for any application including but not limited toaerospace, heavy machinery, and civil engineering (bridges, buildings,etc.) will benefit herefrom.

The spherical elastomeric bearing 30 includes a central bearing element32 having a spherical bearing surface 32 s which defines a bearing focalpoint 30 f. The bearing focal point 30 f defines the flap, lead-lag andpitch axes, Fa, La, and Pa, respectively, about which the rotor bladeassembly articulates (FIG. 1).

To the spherical surface 32 s is bonded discrete spherical elastomericelements 34 about the bearing focal point 30 f. Furthermore, eachspherical elastomeric element 34 includes a multiple of alternatinglayers (see FIG. 2B) of elastomer 36 and nonresilient shims 38,respectively, which are disposed at increasing radii from the bearingfocal point 30 f and have a center of curvature C_(c) which iscoincident therewith.

Each nonresilient shim 38 of the elastomeric spherical bearing 30 istailored to an improved, and in one exemplary embodiment an optimized,thickness as will be further described below. Generally, eachnonresilient shim 38 away from the bearing focal point 30 f increases inthickness. As each nonresilient shim 38 is of an improved, and in anexemplary embodiment optimized, thickness the entire sphericalelastomeric bearing size envelope and weight is improved and may beminimized. The improvement occurs because one or more of thenonresilient shims 38 has a calculated and fabricated size that isdifferent from the thickness of another of the nonresilient shims 38.Optimization occurs, e.g., when each of the thicknesses of each of thenonresilient shims 38 is calculated according to certain criteria asdiscussed below.

Referring to FIG. 3, the elastomeric spherical bearing 30 isschematically illustrated such that nomenclature may be defined. Theelastomeric spherical bearing 30 includes an outwardly facing convexsurface 40 and an inwardly facing concave surface 42. A representativenumber of elastomer layers 36A, 36B, 36C, 36D sandwich a number ofnonresilient shims 38A, 38B, 38C therebetween. It should be understoodthat any number of layers may be included and calculated according tothe disclosure herein. It is noted that superscript or subscript “i”refers to elastomeric layers, while superscript or subscript “j” refersto shim layers.

Symbols (see FIGS. 3, 4, 5, 6A and 6B)

${\hat{T}}_{e}\; = \mspace{11mu} {{Average}\mspace{14mu} {Elastomer}\mspace{14mu} {Thickness}\mspace{14mu} \frac{\left( {T_{i} + T_{i + 1}} \right)}{2}({in})}$

T_(s)=Shim Thickness T_(s) ^(j) (in)

{circumflex over (T)}_(s)=Average Shim Thickness (in)

μ=Shim Poison Ratio

E=Shim Modulus of Elasticity (psi)

R_(s)=Shim Median Spherical Radius R^(j) (in)

$G = \mspace{14mu} {{Average}\mspace{14mu} {Elastomer}\mspace{14mu} {Shear}\mspace{14mu} {Modulus}\mspace{14mu} \frac{\left( {G_{i} + G_{i + 1}} \right)}{2}({psi})}$

B_(I)=Bearing Inner Wrap Angle (deg)

B_(o)=Bearing Outer Wrap Angle (deg)

N_(s)=Total Number of Shims

B_(s)=Outer Wrap Around Angle of Shim B_(s) ^(j) (deg)

D=Shim Bending Stiffness Term (lb-in)

B_(ET)=Shim Geometric Stiffness Term (in)

γ=Average Elastomer Compression Induced Shear Strain (in/in)/100

T_(T)=Total Bearing Elastomer Thickness (in)

Q=Averaging Coefficient of Bearing (non-dimensional)

ν=Averaging Coefficient of Elastomer To Shim Context (non-dimensional)

L_(L)=Unsupported Spherical Arc of Shim L^(i) _(L) (see FIG. 4)

Y_(Y)=Radial Deflection of Shim Y_(Y) ^(i) (see FIG. 4)

σ_(θ)=Shim Hoop Stress σ_(θi) (psi)

D_(P)=Differential Resultant Pressure Load On Unsupported Spherical Arcof Shim L^(i) _(L)

T_(c)=Axial Compressive Load (lb)

θ=Torsional Motion Angle (deg)

β=Cocking Motion Angle (deg)

T_(c)·sin ·β=Radial Load (lb)

R_(I)=Bearing Inner Radius (in)

R_(o)=Bearing Outer Radius (in)

T_(e) ^(i)=Elastomer Thickness (in)

α=Bearing Outer Taper Angle (deg)

Shim Stress Design Module Equations

$\begin{matrix}{R_{R} = {R_{s} \cdot {\sin \left( B_{s} \right)}}} & \lbrack 1\rbrack \\{D = \frac{E \cdot T_{s}^{3}}{12 \cdot \left( {1 - \mu^{2}} \right)}} & \lbrack 2\rbrack \\{B_{ET} = \left\lbrack \frac{E \cdot {T_{s}\left\lbrack {\cos \left( B_{s} \right)} \right\rbrack}^{2}}{4 \cdot D \cdot R_{R}^{2}} \right\rbrack^{1/4}} & \lbrack 3\rbrack \\{D_{P} = {2 \cdot \gamma \cdot G \cdot \frac{L_{L}}{T_{e}}}} & \lbrack 4\rbrack \\{L_{L} = {\left( {B_{I} - B_{o}} \right)\frac{\pi}{180}\left( \frac{R_{s}}{N_{s}} \right){Q \cdot \alpha}}} & \lbrack 5\rbrack \\{Q = \frac{T_{s}}{\left( \frac{T_{T}}{N_{s} + 1} \right)}} & \lbrack 6\rbrack \\{v = \frac{T_{T}}{{N_{s} \cdot \hat{T_{s}}} + T_{T}}} & \lbrack 7\rbrack \\{Y_{Y} = {\frac{D_{P}}{10}\left( \frac{1}{2 \cdot D \cdot B_{ET}^{3}} \right)}} & \lbrack 8\rbrack \\{\sigma_{~\theta} = {Y_{Y}\left\lbrack {\frac{E}{R_{R}}{\cos (B)}} \right\rbrack}} & \lbrack 9\rbrack\end{matrix}$

Equations 1-9 are combined to generate the following equation:

$\begin{matrix}{\sigma_{\theta} = {\frac{4 \cdot \left( {B_{I} - B_{o}} \right) \cdot \gamma \cdot G \cdot D \cdot \pi \cdot E \cdot {\cos \left( B_{s} \right)} \cdot R_{s}}{1800 \cdot T_{e} \cdot R_{s} \cdot {\sin \left( B_{s} \right)} \cdot N_{s}} \cdot \frac{T}{\left( \frac{T_{T}}{N_{s} + 1} \right)} \cdot \frac{T_{T}}{{N_{s} \cdot {\hat{T}}_{s}} + T_{T}} \cdot \left\lbrack \left\lbrack \frac{E \cdot {T_{s}\left\lbrack {\cos \left( B_{s} \right)} \right\rbrack}^{2}}{4 \cdot \frac{E \cdot T_{s}^{3}}{12 \cdot \left( {1 - \mu^{2}} \right)} \cdot \left\lbrack {R_{s} \cdot {\sin \left( B_{s} \right)}} \right\rbrack^{2}} \right\rbrack^{1/4} \right\rbrack^{- 3}}} & \lbrack 10\rbrack\end{matrix}$

Referring to FIG. 7A, utilizing the Equations described above, onecalculation procedure according to one non-limiting embodiment of thepresent invention is as follows:

Calculate Shim Stress

Case I

Compressive Load T_(c) And Torsional Motion (θ) Calculation

Step 1. Determine Loads and Motions.

-   -   a. Determine the axial load. The axial load acts in a direction        parallel to the bearing Z axis of rotation. For rotary-wing        aircraft, this is generally the centrifugal force from the rotor        blade 14.    -   b. Determine the radial load. The radial load acts in the R,        radial direction. This is generally the shear load on the rotor        blade 14.    -   c. Determine the vibratory pitch motion (θ) angle. This motion        rotates about the Z axis and is sometimes referred to as the        torsional rotation. This motion is a vibratory motion. The angle        is a plus or minus motion from the origin.    -   d. Determine the flapping (cocking β) motion angle. (used in        Case II below).    -   e. Determine shim stress allowable (in psi).

Step 2. Enter spherical elastomeric bearing dimensions. Inner radius(R_(I)), Inner Angle (B_(I)), Side Angle (α) sometimes referred to asouter taper angle, and Elastomer modulus of elasticity (E).

Step 3. Use conventional spherical bearing methodology to design firstelastomeric layer 36A (FIG. 4). For further understanding of otheraspects of the conventional spherical bearing methodology, attention isdirected to U.S. patent application Ser. No. 11/959,949 filed Dec. 19,2007 and entitled UNIFORM FATIGUE LIFE SPHERICAL ELASTOMERIC BEARINGwhich is assigned to the assignee of the instant application and whichis hereby incorporated herein in its entirety. Elastomer thickness(T_(e) ^(i)), elastomer shear modulus (G_(e) ^(i)), axial compressioninduced shear stress (γ_(ai)).

Step 4. Assume metal shim thickness (T_(s) ^(j))

Step 5. Repeat Step 3 for the next layer

(T _(e) ^(i+1) , G _(e) ^(i+1), γ_(ai+1))

Step 6. Apply design module equations 1-9 in order to determinenonresilient shim thickness (T_(s) ^(j)) that satisfies materialstrength allowables (e.g., specific to the material being used in theshim).

Step 7. Finalize with existing design module equations 1-9. It is notedthat the calculated nonresilient shim thickness determined in Step 6 maybe modified by addition of another elastomeric layer, and thus Step 7allows a further refinement of the determined nonresilient shimthickness (T_(s) ^(j)) from Step 6.

(T _(e) ^(i+1) , G _(e) ^(i+1), γ_(ai+1))

Step 8. Iterate from Step 4 until all elastomeric layers 36 andnonresilient shims 38 satisfy loads, motions and allowable constraints,e.g., material strength allowables and the constraints described in theaforementioned U.S. patent application Ser. No. 11/959,949 filed Dec.19, 2007 and entitled UNIFORM FATIGUE LIFE SPHERICAL ELASTOMERICBEARING.

Referring to FIG. 7B, utilizing the Equations described above, anothercalculation procedure according to one non-limiting embodiment of thepresent invention is as follows:

Case II

Compressive Load (T_(c)); Torsional Motion (θ); And Cocking Motions (β)Calculation

Follow Case I Module but:

Adjust equation 6 to reflect the change in differential support zone dueto the cocking.

$L_{L} = {\left( {B_{I} - \left( {B_{o} - \beta} \right)} \right)\frac{\pi}{180}\left( \frac{R_{s}}{N_{s}} \right){Q \cdot \upsilon}}$

Adjust equation 7 to reflect the radial loading T_(c)·sin·β due to thecocking motion with associated radial and axial compression inducedshear strain.

$D_{P} = {2 \cdot \left( {\gamma_{a} + \gamma_{r}} \right) \cdot G \cdot \frac{L_{L}}{T_{e}}}$

Modification of equations 6 and 7 incorporate blade flapping in thestress calculation. Follow Case I Design Optimization Module where theshim stress equations are combined to equation 10 until all elastomericlayers and nonresilient shims satisfy load, motions and allowableconstraints.

It should be understood that the instructions are equivalent for a solidbearing and a bearing with a central opening. It should also beunderstood that an elastomeric bearing with any number of layers may becalculated by the method herein.

By way of illustration, an elastomeric bearing 30 with threenonresilient shims 38 according to one non-limiting embodiment, hasinputs delineated in the chart below:

Inputs Bearing Parameters Side Angle, ALPHA (deg) 27 Inner Angle Bi(deg) 54.00 Inner Radius Ri (in) 7.25 Shim Layers Shim Elastic Modulus,E (psi) 3.00E+07 Shim Poisson's Ratio, v 0.3 Shim Hoop Stress Allowable(psi) 10,000 Applied Loads and Motions Axial Loads, Paxial (lbf) 122,500Torsional Rotation, theta (deg) 1.68 Flapping Rotation, beta (deg) 0.96

To provide the following properties:

Final Bearing Properties Geometry Inner Radius Ri (in) 7.25 Outer RadiusRo (in) 7.90 Inner Angle Bi (deg) 54.00 Outer Angle Bo (deg) 51.62 SideAngle, ALPHA (deg) 27 Average Shim Thickness (in) 0.040 Total ElastomerThickness, Tt (in) 0.531 Number of shims, Ns 3 Number of ElastomerLayers 4

These properties provide a nonresilient shim 38A, 38B, 38C whichincrease in thickness by outer layer (FIG. 8) such that eachnonresilient shim 38A, 38B, 38C is subject to an essentially equivalentstress level (FIG. 9).

It should be noted that a computing device can be used to implementvarious functionality and calculations of the design module equations todetermine that each of the nonresilient shims satisfy the desired loads,motions, allowables and other requirements as described herein. In termsof hardware architecture, such a computing device can include aprocessor, memory, and one or more input and/or output (I/O) deviceinterface(s) that are communicatively coupled via a local interface. Thelocal interface can include, for example but not limited to, one or morebuses and/or other wired or wireless connections. The local interfacemay have additional elements, which are omitted for simplicity, such ascontrollers, buffers (caches), drivers, repeaters, and receivers toenable communications. Further, the local interface may include address,control, and/or data connections to enable appropriate communicationsamong the aforementioned components.

The processor may be a hardware device for executing software,particularly software stored in memory. The processor can be a custommade or commercially available processor, a central processing unit(CPU), an auxiliary processor among several processors associated withthe computing device, a semiconductor based microprocessor (in the formof a microchip or chip set) or generally any device for executingsoftware instructions.

The memory can include any one or combination of volatile memoryelements (e.g., random access memory (RAM, such as DRAM, SRAM, SDRAM,VRAM, etc.)) and/or nonvolatile memory elements (e.g., ROM, hard drive,tape, CD-ROM, etc.). Moreover, the memory may incorporate electronic,magnetic, optical, and/or other types of storage media. Note that thememory can also have a distributed architecture, where variouscomponents are situated remotely from one another, but can be accessedby the processor.

The software in the memory may include one or more separate programs,each of which includes an ordered listing of executable instructions forimplementing logical functions. A system component embodied as softwaremay also be construed as a source program, executable program (objectcode), script, or any other entity comprising a set of instructions tobe performed. When constructed as a source program, the program istranslated via a compiler, assembler, interpreter, or the like, whichmay or may not be included within the memory.

The Input/Output devices that may be coupled to system I/O Interface(s)may include input devices, for example but not limited to, a keyboard,mouse, scanner, microphone, camera, proximity device, etc. Further, theInput/Output devices may also include output devices, for example butnot limited to, a printer, display, etc. Finally, the Input/Outputdevices may further include devices that communicate both as inputs andoutputs, for instance but not limited to, a modulator/demodulator(modem; for accessing another device, system, or network), a radiofrequency (RF) or other transceiver, a telephonic interface, a bridge, arouter, etc.

When the computing device is in operation, the processor can beconfigured to execute software stored within the memory, to communicatedata to and from the memory, and to generally control operations of thecomputing device pursuant to the software. Software in memory, in wholeor in part, is read by the processor, perhaps buffered within theprocessor, and then executed.

It should be understood that relative positional terms such as“forward,” “aft,” “upper,” “lower,” “above,” “below,” and the like arewith reference to the normal operational attitude of the vehicle andshould not be considered otherwise limiting.

It should be understood that although a particular component arrangementis disclosed in the illustrated embodiment, other arrangements willbenefit from the instant invention.

Although particular step sequences are shown, described, and claimed, itshould be understood that steps may be performed in any order, separatedor combined unless otherwise indicated and will still benefit from thepresent invention.

The foregoing description is exemplary rather than defined by thelimitations within. Many modifications and variations of the presentinvention are possible in light of the above teachings. The disclosedembodiments of this invention have been disclosed, however, one ofordinary skill in the art would recognize that certain modificationswould come within the scope of this invention. It is, therefore, to beunderstood that within the scope of the appended claims, the inventionmay be practiced otherwise than as specifically described. For thatreason the following claims should be studied to determine the truescope and content of this invention.

1. An elastomeric spherical bearing comprising: a multitude ofelastomeric layers defined about a bearing focal point; and a multitudeof shims, each of said multitude of shims mounted between at least twoof said multiple of elastomeric layers, a first shim of said multitudeof shims having a first thickness and a second shim of said multitude ofshims having a second thickness, said second thickness different thansaid first thickness.
 2. The elastomeric spherical bearing as recited inclaim 1, wherein an inner layer of said multiple of elastomeric layersis mounted to a central bearing element having a spherical bearingsurface.
 3. The elastomeric spherical bearing as recited in claim 2,wherein an inner layer of said multiple of elastomeric layers is mountedto a rotor assembly component.
 4. The elastomeric spherical bearing asrecited in claim 2, wherein an outer layer of said multiple ofelastomeric layers is mounted to a cuff structures of a rotor assembly.5. The elastomeric spherical bearing as recited in claim 1, wherein saidsecond shim is outboard of said first shim relative said bearing focalpoint.
 6. The elastomeric spherical bearing as recited in claim 1,wherein said second thickness is greater than said first thickness. 7.The elastomeric spherical bearing as recited in claim 1, wherein saidmultitude of shims are arranged relative to said bearing focal pointsuch that each of said multitude of shims increase in thickness relativean adjacent inboard shim.
 8. The elastomeric spherical bearing asrecited in claim 1, wherein each of said multitude of shims have agenerally equivalent stress level.
 9. A method of calculating anonresilient shim thickness for a spherical elastomeric bearingcomprising: determining a first thickness of a first of a multitude ofnonresilient shim layers to satisfies a first set of criteria;determining a second thickness of a second of the multitude ofnonresilient shim layers outboard of the first of the multitude ofnonresilient shim layers to satisfies a second set of criteria, thesecond thickness different than the first thickness.
 10. A method asrecited in claim 9, further comprising: determining the first thicknessand the second thickness to generate a generally equivalent stress ineach of the multitude of nonresilient shim layers.
 11. A method asrecited in claim 9, further comprising: utilizing material strengthallowables of each of the multitude of nonresilient shim layers at leastin part as the first criteria and the second criteria.
 12. A method asrecited in claim 9, further comprising: adjusting the shim hoop stressfor each of the multitude of nonresilient shim layers to satisfy:$\sigma_{\theta} = {\frac{4 \cdot \left( {B_{I} - \left( {B_{o} - \beta} \right)} \right) \cdot \left( {\gamma_{a} + \gamma_{r}} \right) \cdot G \cdot D \cdot \pi \cdot E \cdot {\cos \left( B_{s} \right)} \cdot R_{s}}{1800 \cdot T_{e} \cdot R_{s} \cdot {\sin \left( B_{s} \right)} \cdot N_{s}} \cdot \frac{T}{\left( \frac{T_{T}}{N_{s} + 1} \right)} \cdot \frac{T_{T}}{{N_{s} \cdot {\hat{T}}_{s}} + T_{T}} \cdot \left\lbrack \left\lbrack \frac{E \cdot {T_{s}\left\lbrack {\cos \left( B_{s} \right)} \right\rbrack}^{2}}{4 \cdot \frac{E \cdot T_{s}^{3}}{12 \cdot \left( {1 - \mu^{2}} \right)} \cdot \left\lbrack {R_{s} \cdot {\sin \left( B_{s} \right)}} \right\rbrack^{2}} \right\rbrack^{1/4} \right\rbrack^{- 3}}$to produce a spherical elastomeric bearing which generates a generallyequivalent stress to each nonresilient shim layer, where:${\hat{T}}_{e} = {{Average}\mspace{14mu} {Elastomer}\mspace{14mu} {Thickness}\mspace{14mu} \frac{\left( {T_{i} + T_{i + 1}} \right)}{2}({in})}$T_(s)=Shim Thickness T_(s) ^(j) (in) {circumflex over (T)}_(s)=AverageShim Thickness (in) μ=Shim Poison Ratio E=Shim Modulus of Elasticity(psi) R_(s)=Shim Median Spherical Radius R^(j) (in)$G = \; {{Average}\mspace{14mu} {Elastomer}\mspace{14mu} {Shear}\mspace{14mu} {Modulus}\mspace{14mu} \frac{\left( {G_{i} + G_{i + 1}} \right)}{2}({psi})}$B_(I)=Bearing Inner Wrap Angle (deg) B_(o)=Bearing Outer Wrap Angle(deg) N_(s)=Total Number of Shims B_(s)=Outer Wrap Around Angle of ShimB_(s) ^(j) (deg) D=Shim Bending Stiffness Term (lb-in) B_(ET)=ShimGeometric Stiffness Term (in) γ=Average Elastomer Compression InducedShear Strain (in/in)/100 T_(T)=Total Bearing Elastomer Thickness (in)Q=Averaging Coefficient of Bearing (non-dimensional) ν=AveragingCoefficient of Elastomer To Shim Context (non-dimensional)L_(L)=Unsupported Spherical Arc of Shim L^(i) _(L) (see FIG. 4)Y_(Y)=Radial Deflection of Shim Y_(Y) ^(i) (see FIG. 4) σ_(θ)=Shim HoopStress σ_(θi) (psi) D_(P)=Differential Resultant Pressure Load OnUnsupported Spherical Arc of Shim L^(i) _(L) T_(c)=Axial CompressiveLoad (lb) θ=Torsional Motion Angle (deg) β=Cocking Motion Angle (deg)T_(c)·sin·β=Radial Load (lb) R_(I)=Bearing Inner Radius (in)R_(o)=Bearing Outer Radius (in) T_(e) ^(i)=Elastomer Thickness (in)α=Bearing Outer Taper Angle (deg)
 13. A method as recited in claim 9,further comprising: sequentially iterating the thickness of each of themultitude of nonresilient shim layers until the multitude ofnonresilient shim layers to generate a generally equivalent stress ineach of the multitude of nonresilient shim layers.
 14. A method asrecited in claim 9, wherein arranging each of the multitude ofnonresilient shim layers relative to a bearing focal point such thateach of the multitude of nonresilient shim layer increase in thicknessrelative a first inboard nonresilient shim layer.
 15. A method ofcalculating a nonresilient shim layer thickness for a sphericalelastomeric bearing comprising: determining a first shim thickness of afirst of a multitude of nonresilient shim layers in the sphericalelastomeric bearing, each of the multitude of nonresilient shim layerssandwiched between a first pair of elastomeric layers; determining aelastomeric layer thickness for the first pair of elastomeric layers;and modifying the first shim thickness and the elastomeric layerthickness in response to an addition of a second a of the multitude ofnonresilient shim layers and a second pair of elastomeric layersoutboard of the first of the multitude of nonresilient shim layers andthe first pair of elastomeric layers, the modifying continuing untileach of the multitude of nonresilient shim layers have a generallyequivalent stress.
 16. A method as recited in claim 15, furthercomprising: determining an axial load for the spherical elastomericbearing in a direction parallel to a Z axis of rotation; determining aradial load in a radial direction for the spherical elastomeric bearing;and determining a vibratory pitch motion (θ) angle which rotates aboutthe Z axis for the spherical elastomeric bearing.
 17. A method asrecited in claim 18, further comprising: utilizing a centrifugal forcefrom a rotor blade as the axial load; and utilizing a shear load on therotor blade as the radial load.